The Rao–Reiter criterion for the amenability of homogeneous spaces

We prove that a homogeneous space $G/H$, with $G$ a locally compact group and $H$ a closed subgroup of $G$, is amenable in the sense of Eymard–Greenleaf if and only if the quasiregular action $\pi_\Phi$ of $G$ on the unit sphere of the Orlicz space $L^\Phi(G/H)$ for some $N$-function $\Phi\in\Delta_2$ satisfies the Rao–Reiter condition $(P_\Phi)$.